# Generalization of the Wall Theorem to Out-of-equilibrium Conditions

## TL;DR

This paper extends the classical Wall theorem to out-of-equilibrium conditions, providing an exact relation among pressure, density, and temperature at a confining wall, validated through simulations of a flowing nanoconfined liquid.

## Contribution

The authors derive an exact Non-equilibrium Wall theorem applicable to stationary non-equilibrium states, expanding the theorem's applicability beyond equilibrium conditions.

## Key findings

- Analytical relation matches simulation data across various non-equilibrium regimes.
- Excellent agreement between theory and molecular dynamics simulations.
- Assessment of equilibrium Wall theorem accuracy in non-equilibrium scenarios.

## Abstract

The well-known Wall theorem states a simple and precise relation among temperature, pressure and density of a fluid at contact with a confining hard wall in thermodynamic equilibrium. In this Letter we develop an extension of the Wall theorem to out-of-equilibrium conditions, providing an exact relation between pressure, density and temperaure at the wall, valid for strong non-equilibrium situations. We derive analytically this Non-equilibrium Wall theorem for stationary states and validate it with non-equilibrium event-driven molecular-dynamics simulations. We compare the analytical expression with simulations by direct evaluation of temperature, density and pressure on the wall in linear regime, medium and very strong out-of-equilibrium conditions of a nanoconfined liquid under flow in stationary state, presenting viscous heating and heat transport. The agreement between theory and simulation is excellent, allowing for a conclusive validation. In addition, we explore the degree of accuracy of using the equilibrium Wall theorem and different expressions for the local temperature, employed in non-equilibrium molecular-dynamics simulations.